The Mathematics of GeoEnergy

Double-Sideband Suppressed-Carrier Modulation vs Triad

An intriguing yet under-reported finding concerning climate dipole cycles is the symmetry in power spectra observed. This was covered in a post on auto-correlations. The way that this symmetry reveals itself is easily explained by a mirror-folding about one-half some selected carrier frequency, as shown in Fig. 1 below.

Figure 1 : ENSO amplitude spectrum is mirror-folded about 1/2 the annual frequency. Both the data and model align with their mirrored counterparts, as seen in highlighted box.

This works best with a cosine amplitude spectrum, as the sine amplitude spectrum will reveal an anti-symmetry.

Across the next equivalent Brillouin zone — corresponding to the semi-annual harmonic — the mirror symmetry is still there but spotty in terms of alignment as shown in Fig 2.

Figure 2: Mirror symmetry with the range 1.5 to 2 reversed and compared to 1 to 1.5

This mirror symmetry also appears for more obscure climate indices such as the Pacific North America (PNA) pattern as shown in Fig 3 below.

As with ENSO, the PNA essentially characterizes the USA climate variability in ways that a “lot of management policies depend”, governing fisheries, agriculture, irrigation, etc. So even though ENSO and PNA don’t match in terms of time-series comparison, they do share this mirror symmetry characterization.

Davis et al stimulate a partially trapezoidal fluid reservoir with a primary frequency and find that richer harmonics are created with increased forcing as shown in Fig 4 below

Figure 4: Nonlinear production of harmonics from Davis et al with increased forcing

Their interpretation of the experimental results are somewhat inscrutable (IMO) for the time being, but they do call attention to the role of triadic interactions. Triads were discussed in a recent post but the twist here is to make the connection of the mirror folding to the algebraic construction of a triad. In each case shown in Fig 5 below, a peak identified in the Davis presentation shows the same folded mirror symmetry. This reappears across 3 zones as indicated by the arrow annotations.

Figure 5: Annotated triads from Davis et al showing how the triad identity (equation on the right) is equivalent to a mirror folding about 1/2 of the carrier forcing frequency (g). So that the side-band peaks (c) and (f) are mirror folded about (g/2).

The tenuous connection that I am exploring is how the triad frequencies bifurcate with increasing forcing levels. Based on the ENSO model, the mirror symmetry is due to the amplification of the carrier signal — i.e. the annual impulse — against the tidal cycle forcing. This is then fed in to the solution of Laplace’s Tidal Equations, producing a modulated time series rich in additional harmonics. Importantly, this preserves the mirror symmetry.

For comparison, Davis et al generate the chart in Fig 6 to show the progression in richness with increased forcing. Their N is a characteristic buoyancy frequency that figures in to the value of the resonant frequencies.

Figure 6: Increasing forcing from Davis et al

The LTE complement is shown in Fig 7 below where the interaction of a single selected frequency (which could be a resonant frequency such as N) is modulated by the main carrier impulsed wave-train. At the top with a relatively weak LTE modulation of 1, only a single side-band satellite pair is seen for each harmonic zone. With a LTE modulation of 3, another pair of side-band satellites fully emerges, with a third pair beginning to emerge. With modulation of 5 and 7, the strengths of these side-bands begins to equalize across the spectrum. This is all a consequence of higher and thus more energetic standing wavenumbers being added to the system, as a specific value of LTE modulation is linearly related to a standing-wave wavenumber.

Figure 7: This is Fig 6 rotated by 90 degrees, with a superimposed LTE modulation (log scale) of a single frequency amplified by a carrier pulse train.

That’s essentially a demonstration of the DSSC modulation mixing with the LTE modulation to generate the triads — a different yet complementary interpretation that I covered a few weeks ago before attending the EGU sessions.

Here is a link to a recent paper from Davis et al on arXiv that covers their EGU presentation in greater depth : Succession of resonances to achieve internal wave turbulence. This is required reading to remove some of the inscrutability, yet still something seems over-cooked. It’s entirely possible that a natural resonance is interacting with the forcing wave — the fact that the LTE solution requires a single natural response frequency and yet all the satellite side-band pairs (counting 3) clearly align in Fig 7 may not be a coincidence.